Past exam of the options I course of the University of Oxford / 2008 / K / K2 / iii / Solution

For positive wavenumber $k$, the phase velocity is $c_p=\omega /k$, the speed of a fixed phase. The group velocity is $c_g=d\omega /dk$, the leading speed of a narrow wave packet. Differentiating the supplied dispersion relation givesFor fixed $k>0$, the shallow-depth limit uses $\tanh s/s\to 1$: